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Let us not impose any IR cutoff on Minkowski space. The massive scalar's vacuum is given by a normalizable state. However, since there is a continuum of energies for the massless scalar starting from the vacuum itself, is it correct to conclude that the massless scalar's vacuum is non-normalizable? (Note that the continuum of energy states for massive scalar doesn't start from the vacuum)

There might be another way to see this issue. A free particle in non-relativistic QM doesn't have a normalizable vacuum, however as soon as you turn on a potential we get a normalizable vacuum. Does this logic extend to massless scalar's case as well?

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There is no such thing as a "non-normalizable state" in a well-defined theory. The theory of the free scalar field lives on the Fock space which is a specific representation of the CCR on the Hilbert space (see this question for the definition). The Hilbert space is by definition normalizable, so all states in the Fock space are normalizable even when $m=0$.

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  • $\begingroup$ Thanks for your answer. By "well-defined theory", you mean one with an IR cutoff, right? Let's say I don't impose one, then the ground state should be non-normalizable, as in the case of free particle in QM? $\endgroup$
    – Michael Williams
    Mar 7, 2021 at 17:12

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